where \( \phi ^\ast \)  denotes the conjugate norm given by
\(\phi ^\ast (u) = \inf \lbrace  v \bullet u: \phi (v) = 1 \rbrace \)  for \(u \in \mathbf {R}^{n}\) ;
see [1]} and [2]}. In convex and non-smooth Analysis the
distance function and the associated metric projection onto closed subsets
satisfying various convexity-type assumptions (e.g. convexity, positive reach,
proximal smoothness etc..) have been thoroughly investigated both in finite and
infinite dimensional spaces; see [3]}. In Riemannian geometry,
besides more classical and well known applications, the distance function has
recently emerged as a key tool for isoperimetric problems [4]}
and regularity of optimal transport [5]}.
